accelerated sample1 - prek 12 · sample copyright im 2019. im 6–8 math was originally developed...

ACCELERATED7

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Kendall Hunt Publishing Company | im.kendallhunt.com | 1-800-542-6657

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Lakan Cook

Sample copyright IM 2019.

IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and iscopyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0International License (CC BY 4.0), creativecommons.org/licenses/by/4.0/. OUR's 6–8 Math Curriculum is available athttps://openupresources.org/math-curriculum/.

Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics,www.illustrativemathematics.org, and are licensed under the Creative Commons Attribution 4.0 InternationalLicense (CC BY 4.0), creativecommons.org/licenses/by/4.0/.

Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources,openupresources.org, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY4.0), https://creativecommons.org/licenses/by/4.0/.

Adaptations and additions to create the Accelerated version of IM 6-8 Math are copyright 2020 by IllustrativeMathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0)

The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be usedwithout the prior and express written consent of Illustrative Mathematics.

This book includes public domain images or openly licensed images that are copyrighted by their respectiveowners. Openly licensed images remain under the terms of their respective licenses. See the image attributionsection for more information.

The LearnZillion name, logo, and cover artwork are not subject to the Creative Commons license and may not beused without the prior and express written consent of LearnZillion.

Accel6_8

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Rigid Transformations

Lesson 1: Moving in the Plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Lesson 2: Naming the Moves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Lesson 3: Making the Moves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Lesson 4: Coordinate Moves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Lesson 5: Describing Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Properties of Rigid Transformations

Lesson 6: No Bending or Stretching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Lesson 7: Rotation Patterns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Lesson 8: Moves in Parallel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Lesson 9: Composing Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Congruence

Lesson 10: What Is the Same? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Lesson 11: Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Angles in a Triangle

Lesson 12: Alternate Interior Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Lesson 13: Adding the Angles in a Triangle. . . . . . . . . . . . . . . . . . . . . . . . 93

Lesson 14: Parallel Lines and the Angles in a Triangle . . . . . . . . . . . . . . 99

Drawing Polygons with Given Conditions

Lesson 15: Building Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Lesson 16: Triangles with 3 Common Measures . . . . . . . . . . . . . . . . . . 112

Lesson 17: Drawing Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Let's Put It to Work

Lesson 18: Rotate and Tessellate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

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Learning Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

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Lesson 1: Moving in the Plane

Let’s describe ways figures can move in the plane.

1.1: Which One Doesn’t Belong: DiagramsWhich one doesn’t belong?

1.2: Triangle Square DanceYour teacher will give you three pictures. Each shows a different set of dance moves.

1. Arrange the three pictures so you and your partner can both see them right way up.Choose who will start the game.

The starting player mentally chooses A, B, or C and describes the dance to theother player.

The other player identifies which dance is being talked about: A, B, or C.

2. After one round, trade roles. When you have described all three dances, come to anagreement on the words you use to describe the moves in each dance.

3. With your partner, write a description of the moves in each dance.

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Unit 1 Lesson 1: Moving in the Plane 3

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Are you ready for more?

We could think of each dance as a new dance by running it in reverse, starting in the 6thframe and working backwards to the first.

1. Pick a dance and describe in words one of these reversed dances.

2. How do the directions for running your dance in the forward direction and thereverse direction compare?

Lesson 1 Summary

Here are two ways for changing the position of a figure in a plane without changing itsshape or size:

Sliding or shifting the figure withoutturning it. Shifting Figure A to the rightand up puts it in the position of FigureB.

Turning or rotating the figure around apoint. Figure A is rotated around thebottom vertex to create Figure C.

Glossary

vertex

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Lesson 1 Practice Problems1. The six frames show a shape's different positions.

Describe how the shape moves to get from its position in each frame to the next.

2. These five frames show a shape's different positions.


Unit 1 Lesson 1: Moving in the Plane 5

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3. Diego started with this shape.

Diego moves the shape down, turns it 90 degrees clockwise, then moves the shape tothe right. Draw the location of the shape after each move.


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Lesson 2: Naming the Moves

Let’s be more precise about describing moves of figures in the plane.

2.1: A Pair of QuadrilateralsQuadrilateral A can be rotated into the position of Quadrilateral B.

Estimate the angle of rotation.

Unit 1 Lesson 2: Naming the Moves 7

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2.2: How Did You Make That Move?Here is another set of dance moves.

1. Describe each move or say if it is a new move.

a. Frame 1 to Frame 2.

b. Frame 2 to Frame 3.

c. Frame 3 to Frame 4.

d. Frame 4 to Frame 5.

e. Frame 5 to Frame 6.

2. How would you describe the new move?

2.3: Card Sort: MoveYour teacher will give you a set of cards. Sort the cards into categories according to thetype of move they show. Be prepared to describe each category and why it is differentfrom the others.


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Lesson 2 Summary

Here are the moves we have learned about so far:

A translation slides a figure without turning it. Every point in the figure goes thesame distance in the same direction. For example, Figure A was translated down andto the left, as shown by the arrows. Figure B is a translation of Figure A.

A rotation turns a figure about a point, called the center of the rotation. Every pointon the figure goes in a circle around the center and makes the same angle. Therotation can be clockwise, going in the same direction as the hands of a clock,or counterclockwise, going in the other direction. For example, Figure A was rotated

clockwise around its bottom vertex. Figure C is a rotation of Figure A.

A reflection places points on the opposite side of a reflection line. The mirror imageis a backwards copy of the original figure. The reflection line shows where the mirrorshould stand. For example, Figure A was reflected across the dotted line. Figure D is areflection of Figure A.

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We use the word image to describe the new figure created by moving the original figure. Ifone point on the original figure moves to another point on the new figure, we callthem corresponding points.

Glossary

clockwise

counterclockwise

reflection

rotation

translation

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Lesson 2 Practice Problems1. Each of the six cards shows a shape.

a. Which pair of cards shows a shape and its image after a rotation?

b. Which pair of cards shows a shape and its image after a reflection?

2. The five frames show a shape's different positions.



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3. The rectangle seen in Frame 1 is rotated to a new position, seen in Frame 2.

Select all the ways the rectangle could have been rotated to get from Frame 1 toFrame 2.

A. 40 degrees clockwise

B. 40 degrees counterclockwise

C. 90 degrees clockwise

D. 90 degrees counterclockwise

E. 140 degrees clockwise

F. 140 degrees counterclockwise

(From Unit 1, Lesson 1.)


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Lesson 3: Making the Moves

Let's draw and describe translations, rotations, and reflections.

3.1: Notice and Wonder: The Isometric GridWhat do you notice? What do you wonder?

Unit 1 Lesson 3: Making the Moves 13

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3.2: Transformation InformationYour teacher will give you tracing paper to carry out the moves specified. Use , , ,and to indicate vertices in the new figure that correspond to the points , , , andin the original figure.

1. In Figure 1, translate triangle so that goes to .

2. In Figure 2, translate triangle so that goes to .

3. In Figure 3, rotate triangle counterclockwise using center .

4. In Figure 4, reflect triangle using line .


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5. In Figure 5, rotate quadrilateral counterclockwise using center .

6. In Figure 6, rotate quadrilateral clockwise using center .

7. In Figure 7, reflect quadrilateral using line .

8. In Figure 8, translate quadrilateral so that goes to .


The effects of each move can be “undone” by using another move. For example, to undothe effect of translating 3 units to the right, we could translate 3 units to the left. Whatmove undoes each of the following moves?

1. Translate 3 units up

2. Translate 1 unit up and 1 unit to the left

3. Rotate 30 degrees clockwise around a point

4. Reflect across a line


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3.3: A to B to CHere are some figures on an isometric grid.

1. Name a transformation that takes Figure to Figure . Name a transformation thattakes Figure to Figure .

2. What is one sequence of transformations that takes Figure to Figure ? Explainhow you know.


Experiment with some other ways to take Figure to Figure . For example, can you do itwith. . .

No rotations?

No reflections?

No translations?

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Lesson 3 Summary

A move or combination of moves is called a transformation. When we do 1 or moremoves in a row, we often call that a sequence of transformations. When a figure is on agrid, we can use the grid to describe a transformation. We use the word image to describethe figure after a transformation. To distinguish the original figure from its image, points inthe image are sometimes labeled with the same letters as the original figure, but with thesymbol attached, as in (pronounced “A prime”) is the image of after atransformation.

A translation can be described by two points. If a translation moves point to point, it moves the entire figure the same distance and direction as the distance and

direction from to . The distance and direction of a translation can be shown byan arrow.

For example, here is atranslation of quadrilateral

that moves to .

A rotation can be described by an angle and a center. The direction of the angle canbe clockwise or counterclockwise.

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For example, quadrilateralis rotated 60

degrees counterclockwiseusing center . This type ofgrid is called an isometricgrid. The isometric grid ismade up of equilateraltriangles. The angles in thetriangles each measure 60degrees, making theisometric grid convenientfor showing rotations of 60degrees.

A reflection can be described by a line of reflection (the “mirror”). Each point isreflected directly across the line so that it is just as far from the mirror line, but is onthe opposite side.For example, pentagon

is reflected acrossline .

Glossary

image

sequence of transformations

transformation

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Lesson 3 Practice Problems1. Apply each transformation described to Figure A. If you get stuck, try using tracing

paper.

a. A translation which takes to

b. A counterclockwise rotation of A, using center , of 60 degrees

c. A reflection of A across line

2. Here is triangle drawn on a grid.

On the grid, draw a rotationof triangle , atranslation of triangle ,and a reflection of triangle

. Describe clearly howeach was done.


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3. Here is quadrilateral and line .

Draw the image of quadrilateral after reflecting it across line .



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Lesson 4: Coordinate Moves

Let’s transform some figures and see what happens to the coordinates of points.

4.1: Translating CoordinatesSelect all of the translations that take Triangle T to Triangle U. There may be more thanone correct answer.

1. Translate to .

2. Translate to .

3. Translate to .

4. Translate to .

Unit 1 Lesson 4: Coordinate Moves 21

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4.2: Reflecting Points on the Coordinate Plane

1. Here is a list of points

On the coordinate plane:

a. Plot each point and label each with its coordinates.

b. Using the -axis as the line of reflection, plot the image of each point.

c. Label the image of each point with its coordinates.

d. Include a label using a letter. For example, the image of point should belabeled .

2. If the point were reflected using the -axis as the line of reflection, whatwould be the coordinates of the image? What about ? ? Explain howyou know.


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3. The point has coordinates .

a. Without graphing, predict the coordinates of the image of point if pointwere reflected using the -axis as the line of reflection.

b. Check your answer by finding the image of on the graph.

c. Label the image of point as .

d. What are the coordinates of ?

4. Suppose you reflect a point using the -axis as line of reflection. How would youdescribe its image?


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4.3: Transformations of a Segment

Apply each of the following transformations to segment .

1. Rotate segment 90 degrees counterclockwise around center . Label the image ofas . What are the coordinates of ?

2. Rotate segment 90 degrees counterclockwise around center . Label the image ofas . What are the coordinates of ?

3. Rotate segment 90 degrees clockwise around . Label the image of asand the image of as . What are the coordinates of and ?

4. Compare the two 90-degree counterclockwise rotations of segment . What is thesame about the images of these rotations? What is different?


Suppose and are line segments of the same length. Describe a sequence oftransformations that moves to .


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Lesson 4 Summary

We can use coordinates to describe points and find patterns in the coordinates oftransformed points.

We can describe a translation by expressing it as a sequence of horizontal and verticaltranslations. For example, segment is translated right 3 and down 2.

Reflecting a point across an axis changes the sign of one coordinate. For example,reflecting the point whose coordinates are across the -axis changes the signof the -coordinate, making its image the point whose coordinates are . Reflectingthe point across the -axis changes the sign of the -coordinate, making the image thepoint whose coordinates are .

Reflections across other lines are more complex to describe.


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We don’t have the tools yet to describe rotations in terms of coordinates in general. Hereis an example of a rotation with center in a counterclockwise direction.

Point has coordinates . Segment was rotated counterclockwise around .Point with coordinates rotates to point whose coordinates are .

Glossary

coordinate plane•


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Lesson 4 Practice Problems1. a. Here are some points.

What are the coordinates of , , and after a translation to the right by 4 unitsand up 1 unit? Plot these points on the grid, and label them , and .

b. Here are some points.

What are the coordinates of , , and after a reflection over the axis? Plotthese points on the grid, and label them , and .


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c. Here are some points.

What are the coordinates of , , and after a rotation about by 90degrees clockwise? Plot these points on the grid, and label them , and .

2. Describe a sequence of transformations that takes trapezoid A to trapezoid B.


3. Reflect polygon using line .



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Lesson 5: Describing Transformations

Let’s transform some polygons in the coordinate plane.

5.1: Finding a Center of RotationAndre performs a 90-degree counterclockwise rotation of Polygon P and gets Polygon P’,but he does not say what the center of the rotation is. Can you find the center?

Unit 1 Lesson 5: Describing Transformations 29

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5.2: Info Gap: Transformation InformationYour teacher will give you either a problem card or a data card. Do not show or read yourcard to your partner.

If your teacher gives you the problem card:

1. Silently read your card and think aboutwhat information you need to be ableto answer the question.

2. Ask your partner for the specificinformation that you need.

3. Explain how you are using theinformation to solve the problem.

Continue to ask questions until youhave enough information to solve theproblem.

4. Share the problem card and solve theproblem independently.

5. Read the data card and discuss yourreasoning.

If your teacher gives you the data card:

1. Silently read your card.

2. Ask your partner “What specificinformation do you need?” and wait forthem to ask for information.

If your partner asks for information thatis not on the card, do not do thecalculations for them. Tell them youdon’t have that information.

3. Before sharing the information, ask“Why do you need that information?”Listen to your partner’s reasoning andask clarifying questions.

4. Read the problem card and solve theproblem independently.

5. Share the data card and discuss yourreasoning.

Pause here so your teacher can review your work. Ask your teacher for a new set of cardsand repeat the activity, trading roles with your partner.


Sometimes two transformations, one performed after the other, have a nice description asa single transformation. For example, instead of translating 2 units up followed bytranslating 3 units up, we could simply translate 5 units up. Instead of rotating 20 degreescounterclockwise around the origin followed by rotating 80 degrees clockwise around theorigin, we could simply rotate 60 degrees clockwise around the origin.

Can you find a simple description of reflecting across the -axis followed by reflectingacross the -axis?


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Lesson 5 Summary

The center of a rotation for a figure doesn’t have to be one of the points on the figure. Tofind a center of rotation, look for a point that is the same distance from two correspondingpoints. You will probably have to do this for a couple of different pairs of correspondingpoints to nail it down.

When we perform a sequenceof transformations, the orderof the transformations can beimportant. Here is triangle

translated up two unitsand then reflected over the

-axis.

Here is triangle reflectedover the -axis and thentranslated up two units.

Triangle ends up indifferent places when thetransformations are appliedin the opposite order!


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Lesson 5 Practice Problems1. Here is Trapezoid A in the coordinate plane:

a. Draw Polygon B, the image of A, using the -axis as the line of reflection.

b. Draw Polygon C, the image of B, using the -axis as the line of reflection.

c. Draw Polygon D, the image of C, using the -axis as the line of reflection.

2. The point is rotated 180 degrees counterclockwise using center . Whatare the coordinates of the image?

A.

B.

C.

D.


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3. Describe a sequence of transformations for which Triangle B is the image of TriangleA.

4. Here is quadrilateral .

Draw the image of quadrilateral after each transformation.

a. The translation that takes to .

b. The reflection over segment .

c. The rotation about point by angle , counterclockwise.



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Lesson 6: No Bending or Stretching

Let’s compare measurements before and after translations, rotations, and reflections.

6.1: Measuring SegmentsFor each question, the unit is represented by the large tick marks with whole numbers.

1. Find the length of this segment to the nearest of a unit.

2. Find the length of this segment to the nearest 0.1 of a unit.

3. Estimate the length of this segment to the nearest of a unit.

4. Estimate the length of the segment in the prior question to the nearest 0.1 of a unit.


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6.2: Sides and Angles1. Translate Polygon so point goes to point . In the image, write the length of each

side, in grid units, next to the side.

2. Rotate Triangle 90 degrees clockwise using as the center of rotation. In theimage, write the measure of each angle in its interior.

Unit 1 Lesson 6: No Bending or Stretching 35

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3. Reflect Pentagon across line .a. In the image, write the length of each side, in grid units, next to the side. You

may need to make your own ruler with tracing paper or a blank index card.

b. In the image, write the measure of each angle in the interior.

6.3: Which One?Here is a grid showing triangle and two other triangles.

You can use a rigid transformation to take triangle to one of the other triangles.

1. Which one? Explain how you know.

2. Describe a rigid transformation that takes to the triangle you selected.


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A square is made up of an L-shaped regionand three transformations of the region. Ifthe perimeter of the square is 40 units, whatis the perimeter of each L-shaped region?


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Lesson 6 Summary

The transformations we’ve learned about so far, translations, rotations, reflections, andsequences of these motions, are all examples of rigid transformations. A rigidtransformation is a move that doesn’t change measurements on any figure.

Earlier, we learned that a figure and its image have corresponding points. With a rigidtransformation, figures like polygons also have corresponding sides and correspondingangles. These corresponding parts have the same measurements.

For example, triangle was made by reflecting triangle across a horizontal line,then translating. Corresponding sides have the same lengths, and corresponding angleshave the same measures.

measurements in triangle corresponding measurements in image

Glossary

corresponding

rigid transformation••


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Lesson 6 Practice Problems1. Is there a rigid transformation taking Rhombus P to Rhombus Q? Explain how you

know.

2. Describe a rigid transformation that takes Triangle A to Triangle B.

3. Is there a rigid transformation taking Rectangle A to Rectangle B? Explain how youknow.


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4. For each shape, draw its image after performing the transformation. If you get stuck,consider using tracing paper.

a. Translate the shape sothat goes to .

a. Rotate the shape 180degreescounterclockwisearound .

a. Reflect the shape overthe line shown.



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Lesson 7: Rotation Patterns

Let’s rotate figures in a plane.

7.1: Building a QuadrilateralHere is a right isosceles triangle:

1. Rotate triangle 90 degrees clockwise around .

2. Rotate triangle 180 degrees clockwise round .

3. Rotate triangle 270 degrees clockwise around .

4. What would it look like when you rotate the four triangles 90 degrees clockwisearound ? 180 degrees? 270 degrees clockwise?

Unit 1 Lesson 7: Rotation Patterns 41

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7.2: Rotating a Segment

1. Rotate segment 180 degrees around point . Draw its image and label the imageof as

2. Rotate segment 180 degrees around point . Draw its image and label the imageof as and the image of as .

3. Rotate segment 180 degrees around its midpoint, What is the image of ?

4. What happens when you rotate a segment 180 degrees around a point?


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Here are two line segments. Is it possible to rotate one line segment to the other? If so,find the center of such a rotation. If not, explain why not.


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7.3: A Pattern of Four TrianglesYou can use rigidtransformations of afigure to makepatterns. Here is adiagram built withthree differenttransformations oftriangle .

1. Describe a rigid transformation that takes triangle to triangle .



4. Do segments , , , and all have the same length? Explain your reasoning.


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Lesson 7 Summary

When we apply a 180-degree rotation to a line segment, there are several possibleoutcomes:

The segment maps to itself (if the center of rotation is the midpoint of the segment).

The image of the segment overlaps with the segment and lies on the same line (if thecenter of rotation is a point on the segment).

The image of the segment does not overlap with the segment (if the center ofrotation is not on the segment).

We can also build patterns by rotating a shape. For example, triangle shown here has. If we rotate triangle 60 degrees, 120 degrees, 180 degrees, 240 degrees,

and 300 degrees clockwise, we can build a hexagon.

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Lesson 7 Practice Problems1. For the figure shown here,

a. Rotate segmentaround point .

b. Rotate segmentaround point .

c. Rotate segmentaround point .

2. Here is an isosceles right triangle:

Draw these three rotationsof triangle together.

a. Rotate triangle 90degrees clockwisearound .

b. Rotate triangle180 degrees around .

c. Rotate triangle270 degrees clockwisearound .


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3. Each graph shows two polygons and . In each case, describe asequence of transformations that takes to .

a.

b.



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4. Lin says that she can map Polygon A to Polygon B using only reflections. Do you agreewith Lin? Explain your reasoning.



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Lesson 8: Moves in Parallel

Let’s transform some lines.

8.1: Line MovesFor each diagram, describe a translation, rotation, or reflection that takes line to line

. Then plot and label and , the images of and .

1.

2.

Unit 1 Lesson 8: Moves in Parallel 49

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8.2: Parallel Lines

Use a piece of tracing paper to trace lines and and point . Then use that tracing paperto draw the images of the lines under the three different transformations listed.

As you perform each transformation, think about the question:

What is the image of two parallel lines under a rigid transformation?

1. Translate lines and 3 units up and 2 units to the right.

a. What do you notice about the changes that occur to lines and after thetranslation?

b. What is the same in the original and the image?


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2. Rotate lines and counterclockwise 180 degrees using as the center of rotation.

a. What do you notice about the changes that occur to lines and after therotation?


3. Reflect lines and across line .

a. What do you notice about the changes that occur to lines and after thereflection?



When you rotate two parallel lines, sometimes the two original lines intersect their imagesand form a quadrilateral. What is the most specific thing you can say about thisquadrilateral? Can it be a square? A rhombus? A rectangle that isn’t a square? Explain yourreasoning.


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8.3: Let’s Do Some 180’s1. The diagram shows a line with points labeled , , , and .

a. On the diagram, draw the image of the line and points , , and after the linehas been rotated 180 degrees around point .

b. Label the images of the points , , and .

c. What is the order of all seven points? Explain or show your reasoning.

2. The diagram shows a line with points and on the line and a segment whereis not on the line.

a. Rotate the figure 180 degrees about point . Label the image of as and theimage of as .

b. What do you know about the relationship between angle and angle? Explain or show your reasoning.


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3. The diagram shows two lines and that intersect at a point with point onand point on .

a. Rotate the figure 180 degrees around . Label the image of as and theimage of as .

b. What do you know about the relationship between the angles in the figure?Explain or show your reasoning.

Lesson 8 Summary

Rigid transformations have the following properties:

A rigid transformation of a line is a line.

A rigid transformation of two parallel lines results in two parallel lines that are thesame distance apart as the original two lines.

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Sometimes, a rigid transformation takes a line to itself. For example:

A translation parallel to the line. The arrow shows a translation of line thatwill take to itself.

A rotation by around any point on the line. A rotation of linearound point will take to itself.

A reflection across any line perpendicular to the line. A reflection of lineacross the dashed line will take to itself.

These facts let us make an important conclusion. If two lines intersect at a point, whichwe’ll call , then a rotation of the lines with center shows that vertical angles arecongruent. Here is an example:

Rotating both lines by around sends angle to angle , proving that theyhave the same measure. The rotation also sends angle to angle .

Glossary

vertical angles

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Lesson 8 Practice Problems1. a. Draw parallel lines and .

b. Pick any point . Rotate 90 degrees clockwise around .

c. Rotate line 90 degrees clockwise around .

d. What do you notice?

2. Use the diagram to find the measures of each angle. Explain your reasoning.

a.

b.

c.


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3. Points and are plotted on a line.

a. Find a point so that a 180-degree rotation withcenter sends to and to .

b. Is there more than one point that works for parta?

4. In the picture triangle is an image of triangle after a rotation. The centerof rotation is .

a. What is the length of side ? Explain how you know.

b. What is the measure of angle ? Explain how you know.

c. What is the measure of angle ? Explain how you know.



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5. The point is rotated 180 degrees counterclockwise using center . Whatare the coordinates of the image?

A.

B.

C.

D.



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Lesson 9: Composing Figures

Let’s use reasoning about rigid transformations to find measurements without measuring.

9.1: Angles of an Isosceles TriangleHere is a triangle.

1. Reflect triangle over line . Labelthe image of as .

2. Rotate triangle around so thatmatches up with .

3. What can you say about the measuresof angles and ?

9.2: Triangle Plus OneHere is triangle .

1. Draw midpoint of side .

2. Rotate triangle 180 degrees usingcenter to form triangle . Drawand label this triangle.

3. What kind of quadrilateral is ?Explain how you know.


In the activity, we made a parallelogram by taking a triangle and its image under a180-degree rotation around the midpoint of a side. This picture helps you justify awell-known formula for the area of a triangle. What is the formula and how does the figurehelp justify it?


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9.3: Triangle Plus TwoThe picture shows 3 triangles. Triangle 2 and Triangle 3 are images of Triangle 1 underrigid transformations.

1. Describe a rigid transformation that takes Triangle 1 to Triangle 2. What points inTriangle 2 correspond to points , , and in the original triangle?

2. Describe a rigid transformation that takes Triangle 1 to Triangle 3. What points inTriangle 3 correspond to points , , and in the original triangle?

3. Find two pairs of line segments in the diagram that are the same length, and explainhow you know they are the same length.

4. Find two pairs of angles in the diagram that have the same measure, and explain howyou know they have the same measure.

Unit 1 Lesson 9: Composing Figures 59

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9.4: Triangle ONE PlusHere is isosceles triangle . Its sidesand have equal lengths. Angle is 30degrees. The length of is 5 units.

1. Reflect triangle across segment . Label the new vertex .

2. What is the measure of angle ?


4. Reflect triangle across segment . Label the point that corresponds to as.

5. How long is ? How do you know?


7. If you continue to reflect each new triangle this way to make a pattern, what will thepattern look like?


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Lesson 9 Summary

Earlier, we learned that if we apply a sequence of rigid transformations to a figure, thencorresponding sides have equal length and corresponding angles have equal measure.These facts let us figure out things without having to measure them!

For example, here is triangle .

We can reflect triangle across side to form a new triangle:

Because points and are on the line of reflection, they do not move. So the image oftriangle is . We also know that:

Angle measures because it is the image of angle .

Segment has the same length as segment .

When we construct figures using copies of a figure made with rigid transformations, weknow that the measures of the images of segments and angles will be equal to themeasures of the original segments and angles.

••


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Lesson 9 Practice Problems1. Here is the design for the flag of Trinidad and Tobago.

Describe a sequence of translations, rotations, and reflections that take the lower lefttriangle to the upper right triangle.

2. Here is a picture of an older version of the flag of Great Britain. There is a rigidtransformation that takes Triangle 1 to Triangle 2, another that takes Triangle 1 toTriangle 3, and another that takes Triangle 1 to Triangle 4.

a. Measure the lengths of the sides in Triangles 1 and 2. What do you notice?

b. What are the side lengths of Triangle 3? Explain how you know.

c. Do all eight triangles in the flag have the same area? Explain how you know.


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3. a. Which of the lines in the picture is parallel to line ? Explain how you know.

b. Explain how to translate, rotate or reflect line to obtain line .

c. Explain how to translate, rotate or reflect line to obtain line .


4. Point has coordinates . After a translation 4 units left, a reflection across the-axis, and a translation 2 units down, what are the coordinates of the image?



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5. Here is triangle :

Draw these three rotations of triangle together.

a. Rotate triangle 90 degrees clockwise around .

b. Rotate triangle 180 degrees around .

c. Rotate triangle 270 degrees clockwise around .



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Lesson 10: What Is the Same?

Let’s decide whether shapes are the same.

10.1: Find the Right HandsA person’s hands are mirror images of each other. In the diagram, a left hand is labeled.Shade all of the right hands.

Unit 1 Lesson 10: What Is the Same? 65

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10.2: Are They the Same?For each pair of shapes, decide whether or not they are the same.


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10.3: Area, Perimeter, and Congruence

1. Which of these rectangles have the same area as Rectangle R but differentperimeter?

2. Which rectangles have the same perimeter but different area?

3. Which have the same area and the same perimeter?

4. Use materials from the geometry tool kit to decide which rectangles are congruent.Shade congruent rectangles with the same color.


In square , points , , , and aremidpoints of their respective sides. Whatfraction of square is shaded? Explainyour reasoning.


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Lesson 10 Summary

Congruent is a new term for an idea we have already been using. We say that two figuresare congruent if one can be lined up exactly with the other by a sequence of rigidtransformations. For example, triangle is congruent to triangle because theycan be matched up by reflecting triangle across followed by the translationshown by the arrow. Notice that all corresponding angles and side lengths are equal.

Here are some other facts about congruent figures:

We don’t need to check all the measurements to prove two figures are congruent; wejust have to find a sequence of rigid transformations that match up the figures.

A figure that looks like a mirror image of another figure can be congruent to it. Thismeans there must be a reflection in the sequence of transformations that matchesup the figures.

Since two congruent polygons have the same area and the same perimeter, one wayto show that two polygons are not congruent is to show that they have a differentperimeter or area.

Glossary

congruent

•

•

•

•


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Lesson 10 Practice Problems1. If two rectangles have the same perimeter, do they have to be congruent? Explain

how you know.

2. Draw two rectangles that have the same area, but are not congruent.

3. For each pair of shapes, decide whether or not the two shapes are congruent. Explainyour reasoning.

a.

b.


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4. a. Reflect Quadrilateral A over the -axis. Label the image quadrilateral B. ReflectQuadrilateral B over the -axis. Label the image C.

b. Are Quadrilaterals A and C congruent? Explain how you know.

5. The point is rotated 90 degrees counterclockwise using center . What arethe coordinates of the image?

A.

B.

C.

D.



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6. Describe a rigid transformation that takes Polygon A to Polygon B.



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Lesson 11: Congruence

Let’s decide if two figures are congruent.

11.1: Translated ImagesAll of these trianglesare congruent.Sometimes we cantake one figure toanother with atranslation. Shade thetriangles that areimages of triangle

under atranslation.


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11.2: Congruent PairsFor each of the following pairs of shapes, decide whether or not they are congruent.Explain your reasoning.

1.

2.

Unit 1 Lesson 11: Congruence 73

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3.

4.


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5.


A polygon has 8 sides: five of length 1, two of length 2, and one of length 3. All sides lie ongrid lines. (It may be helpful to use graph paper when working on this problem.)

1. Find a polygon with these properties.

2. Is there a second polygon, not congruent to your first, with these properties?


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11.3: Corresponding Points in Congruent FiguresHere are two congruentshapes with somecorresponding points labeled.

1. Draw the points corresponding to , , and , and label them , , and .

2. Draw line segments and and measure them. Do the same for segmentsand and for segments and . What do you notice?

3. Do you think there could be a pair of corresponding segments with different lengths?Explain.


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Lesson 11 Summary

How do we know if two figures are congruent?

If we copy 1 figure on tracing paper and move the paper so the copy covers the otherfigure exactly, then that suggests they are congruent.

We can prove that 2 figures are congruent by describing a sequence of translations,rotations, and reflections that move 1 figure onto the other so they match up exactly.

Distances between corresponding points on congruent figures are always equal, evenfor curved shapes. For example, corresponding segments and on thesecongruent ovals have the same length:

How do we know that 2 figures are not congruent?

If there is no correspondence between the figures where the parts have equalmeasure, that proves that the two figures are not congruent. In particular,

If two polygons have different sets of side lengths, they can’t be congruent. Forexample, the figure on the left has side lengths 3, 2, 1, 1, 2, 1. The figure on theright has side lengths 3, 3, 1, 2, 2, 1. There is no way to make a correspondencebetween them where all corresponding sides have the same length.

•

•

•

•

◦


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If two polygons have the same side lengths, but their orders can’t be matchedas you go around each polygon, the polygons can’t be congruent. For example,rectangle can’t be congruent to quadrilateral . Even though theyboth have two sides of length 3 and two sides of length 5, they don’t correspondin the same order. In , the order is 3, 5, 3, 5 or 5, 3, 5, 3; in , theorder is 3, 3, 5, 5 or 3, 5, 5, 3 or 5, 5, 3, 3.

If two polygons have the same side lengths, in the same order, but differentcorresponding angles, the polygons can’t be congruent. For example,parallelogram can’t be congruent to rectangle . Even though theyhave the same side lengths in the same order, the angles are different. Allangles in are right angles. In , angles and are less than 90degrees and angles and are more than 90 degrees.

If you have curved figures, like these 2 ovals, you can find parts of the figuresthat should correspond but that have different measurements. On both, thelongest distance from left to right is 5 units across, and the longest distancefrom top to bottom is 4 units. The line segment from the highest to lowest pointis in the middle of the left oval, but in the right oval, it’s 2 units from the rightend and 3 units from the left end. This proves they are not congruent.

◦

◦

◦


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Glossary

right angle•


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Lesson 11 Practice Problems1. a. Show that the two pentagons are congruent.

b. Find the side lengths of and the angle measures of .

2. These two figures are congruent, with corresponding points marked.

a. Are angles and congruent? Explain your reasoning.

b. Measure angles and to check your answer.


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3. For each pair of shapes, decide whether or not the two shapes are congruent. Explainyour reasoning.

a.

b.

c.


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4. Which of these four figures are congruent to the top figure?

A. A

B. B

C. C

D. D

5. Here are two figures.

Show, using measurement, that these two figures are not congruent.


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Lesson 12: Alternate Interior Angles

Let’s explore why some angles are always equal.

12.1: Angle Pairs1. Find the measure of angle . Explain or show your reasoning.

2. Find and label a second angle in the diagram. Find and label an angle congruentto angle .

3. Angle is a right angle. Find the measure of angle .

Unit 1 Lesson 12: Alternate Interior Angles 83

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12.2: Cutting Parallel Lines with a TransversalLines and are parallel. They are cut by transversal .

1. With your partner, find the seven unknown angle measures in the diagram. Explainyour reasoning.

2. What do you notice about the angles with vertex and the angles with vertex ?


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3. Using what you noticed, find the measures of the four angles at point in the seconddiagram. Lines and are parallel.


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4. The next diagram resemblesthe first one, but the linesform slightly differentangles. Work with yourpartner to find thesix unknown angles withvertices at points and .

5. What do you notice about the angles in this diagram as compared to the earlierdiagram? How are the two diagrams different? How are they the same?


Parallel lines and are cut by two transversals which intersect in the same point. Twoangles are marked in the figure. Find the measure of the third angle.


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12.3: Alternate Interior Angles Are Congruent1. Lines and are parallel and is a transversal. Point is the midpoint of segment

.

Find a rigid transformation showing that angles and are congruent.

2. In this picture, lines and are no longer parallel. is still the midpoint of segment.

Does your argument in theearlier problem apply in thissituation? Explain.


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12.4: Info Gap: Angle FindingYour teacher will give you either a problem card or a data card. Do not show or read yourcard to your partner.

If your teacher gives you the problem card:

1. Silently read your card and think aboutwhat information you need to be ableto answer the question.

2. Ask your partner for the specificinformation that you need.

3. Explain how you are using theinformation to solve the problem.

Continue to ask questions until youhave enough information to solve theproblem.

4. Share the problem card and solve theproblem independently.

5. Read the data card and discuss yourreasoning.

If your teacher gives you the data card:

1. Silently read your card.

2. Ask your partner “What specificinformation do you need?” and wait forthem to ask for information.

If your partner asks for information thatis not on the card, do not do thecalculations for them. Tell them youdon’t have that information.

3. Before sharing the information, ask“Why do you need that information?”Listen to your partner’s reasoning andask clarifying questions.

4. Read the problem card and solve theproblem independently.

5. Share the data card and discuss yourreasoning.

Pause here so your teacher can review your work. Ask your teacher for a new set of cardsand repeat the activity, trading roles with your partner.


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Lesson 12 Summary

If two angle measures add up to , then we say the anglesare complementary. If two angle measures add up to ,then we say the angles are supplementary. When two linesintersect, vertical angles are equal and adjacent angles aresupplementary. For example, in this figure angles 1 and 3are equal, angles 2 and 4 are equal, angles 1 and 4 aresupplementary, and angles 2 and 3 are supplementary.

When two parallel lines are cut by another line, called atransversal, two pairs of alternate interior angles arecreated. (“Interior” means on the inside, or between, the twoparallel lines.) For example, in this figure angles 3 and 5 arealternate interior angles and angles 4 and 6 are alsoalternate interior angles.

Alternate interior angles are equal because a rotation around the midpoint of thesegment that joins their vertices takes each angle to the other. Imagine a point halfwaybetween the two intersections—can you see how rotating about takes angle 3 toangle 5?

Using what we know about vertical angles, adjacent angles, and alternate interior angles,we can find the measures of any of the eight angles created by a transversal if we knowjust one of them. For example, starting with the fact that angle 1 is we use verticalangles to see that angle 3 is , then we use alternate interior angles to see that angle 5 is

, then we use the fact that angle 5 is supplementary to angle 8 to see that angle 8 issince . It turns out that there are only two different measures. In this

example, angles 1, 3, 5, and 7 measure , and angles 2, 4, 6, and 8 measure .

Glossary

alternate interior angles

complementary

supplementary

transversal

••••


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Lesson 12 Practice Problems1. Segments , , and intersect at point , and angle is a right angle. Find

the value of .

2. is a point on line segment . is a line segment. Select all the equations thatrepresent the relationship between the measures of the angles in the figure.

A.

B.

C.

D.

E.

F.


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3. Use the diagram to find the measure of each angle.

a.

b.

c.


4. Lines and are parallel, and the measure of angle is 19 degrees.

a. Explain why the measure of angle is 19 degrees. If you get stuck, considertranslating line by moving to .

b. What is the measure of angle ? Explain.


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5. The diagram shows three lines with some marked angle measures.

Find the missing anglemeasures marked withquestion marks.

6. Lines and are parallel. Find the value of .


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Lesson 13: Adding the Angles in a Triangle

Let’s explore angles in triangles.

13.1: Can You Draw It?1. Complete the table by drawing a triangle in each cell that has the properties listed for

its column and row. If you think you cannot draw a triangle with those properties,write “impossible” in the cell.

2. Share your drawings with a partner. Discuss your thinking. If you disagree, work toreach an agreement.

acute (all anglesacute)

right (has a rightangle)

obtuse (has anobtuse angle)

scalene (sidelengths alldifferent)

isosceles (at leasttwo side

lengthsare equal)

equilateral (threeside lengths equal)

Unit 1 Lesson 13: Adding the Angles in a Triangle 93

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13.2: Find All ThreeYour teacher will give you a card with a picture of a triangle.

1. The measurement of one of the angles is labeled. Mentally estimate the measures ofthe other two angles.

2. Find two other students with triangles congruent to yours but with a different anglelabeled. Confirm that the triangles are congruent, that each card has a different anglelabeled, and that the angle measures make sense.

3. Enter the three angle measures for your triangle on the table your teacher hasposted.

13.3: Tear It UpYour teacher will give you a page with three sets of angles and a blank space. Cut out eachset of three angles. Can you make a triangle from each set that has these same threeangles?


1. Draw a quadrilateral. Cut it out, tear off its angles, and line them up. What do younotice?

2. Repeat this for several more quadrilaterals. Do you have a conjecture about theangles?


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Lesson 13 Summary

A angle is called a straight angle because when it is made with two rays, they point inopposite directions and form a straight line.

If we experiment with angles in a triangle, we find that the sum of the measures of thethree angles in each triangle is —the same as a straight angle!

Through experimentation we find:

If we add the three angles of a triangle physically by cutting them off and lining upthe vertices and sides, then the three angles form a straight angle.

If we have a line and two rays that form three angles added to make a straight angle,then there is a triangle with these three angles.

Glossary

straight angle

•

•

•


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Lesson 13 Practice Problems1. In triangle , the measure of angle is .

a. Give possible measures for angles and if triangle is isosceles.

b. Give possible measures for angles and if triangle is right.

2. For each set of angles, decide if there is a triangle whose angles have these measuresin degrees:

a. 60, 60, 60

b. 90, 90, 45

c. 30, 40, 50

d. 90, 45, 45

e. 120, 30, 30

If you get stuck, consider making a line segment. Then use a protractor to measureangles with the first two angle measures.

3. Angle in triangle is obtuse. Can angle or angle be obtuse? Explain yourreasoning.


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4. For each pair of polygons, describe the transformation that could be applied toPolygon A to get Polygon B.

a.

b.

c.



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5. On the grid, draw a scaled copy of quadrilateral using a scale factor of .



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Lesson 14: Parallel Lines and the Angles in aTriangle

Let’s see why the angles in a triangle add to 180 degrees.

14.1: True or False: Computational RelationshipsIs each equation true or false?

14.2: Angle Plus TwoHere is triangle .

1. Rotate triangle around the midpoint of side . Label the new vertex .

2. Rotate triangle around the midpoint of side . Label the new vertex .

3. Look at angles , , and . Without measuring, write what you think is thesum of the measures of these angles. Explain or show your reasoning.

Unit 1 Lesson 14 99

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4. Is the measure of angle equal to the measure of any angle in triangle ? If so,which one? If not, how do you know?

5. Is the measure of angle equal to the measure of any angle in triangle ? Ifso, which one? If not, how do you know?

6. What is the sum of the measures of angles , , and ?

14.3: Every Triangle in the WorldHere is . Line is parallel to line .

1. What is ? Explain how you know.

2. Use your answer to explain why .

3. Explain why your argument will work for any triangle: that is, explain why the sum ofthe angle measures in any triangle is .


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1. Using a ruler, create a few quadrilaterals. Use a protractor to measure the fourangles inside the quadrilateral. What is the sum of these four angle measures?

2. Come up with an explanation for why anything you notice must be true (hint: drawone diagonal in each quadrilateral).

14.4: Four Triangles RevisitedThis diagram shows a square that has been made by images of triangle underrigid transformations.

Given that angle measures 53 degrees, find as many other angle measures as youcan.

Unit 1 Lesson 14 101

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Lesson 14 Summary

Using parallel lines and rotations, we can understand why the angles in a triangle alwaysadd to . Here is triangle . Line is parallel to and contains .

A 180 degree rotation of triangle around the midpoint of interchanges anglesand so they have the same measure: in the picture these angles are marked as . A180 degree rotation of triangle around the midpoint of interchanges anglesand so they have the same measure: in the picture, these angles are marked as .Also, is a straight line because 180 degree rotations take lines to parallel lines. So thethree angles with vertex make a line and they add up to ( ). Butare the measures of the three angles in so the sum of the angles in a triangle isalways !


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Lesson 14 Practice Problems1. For each triangle, find the measure of the missing angle.

2. Is there a triangle with two right angles? Explain your reasoning.


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3. In this diagram, lines and are parallel.

Angle measures and angle measures .

a. What is ?

b. What is ?

c. What is ?

4. Here is a diagram of triangle .

a. Find the measures ofangles , , and .

b. Find the sum of themeasures of angles ,, and .

c. What do you noticeabout these threeangles?


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5. The two figures are congruent.

a. Label the points , and that correspond to , , and in the figure onthe right.

b. If segment measures 2 cm, how long is segment ? Explain.

c. The point is shown in addition to and . How can you find the point thatcorresponds to ? Explain your reasoning.



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Lesson 15: Building Polygons

Let’s build shapes.

15.1: Where Is Lin?At a park, the slide is 5 meters east of the swings. Lin is standing 3 meters away from theslide.

1. Draw a diagram of the situation including a place where Lin could be.

2. How far away from the swings is Lin in your diagram?

3. Where are some other places Lin could be?

15.2: Building Diego’s and Jada’s Shapes1. Diego built a quadrilateral using side lengths of 4 in, 5 in, 6 in, and 9 in.

a. Build such a shape.

b. Is your shape an identical copy of Diego’s shape? Explain your reasoning.

2. Jada built a triangle using side lengths of 4 in, 5 in, and 8 in.

a. Build such a shape.

b. Is your shape an identical copy of Jada’s shape? Explain your reasoning.


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15.3: Swinging the Sides AroundWe’ll explore a method for drawing a triangle that has three specific side lengths. Yourteacher will give you a piece of paper showing a 4-inch segment as well as someinstructions for which strips to use and how to connect them.

1. Follow these instructions to mark the possible endpoints of one side:

a. Put your 4-inch strip directly on top of the 4-inch segment on the piece of paper.Hold it in place.

b. For now, ignore the 3-inch strip on the left side. Rotate it so that it is out of theway.

c. In the 3-inch strip on the right side, put the tip of your pencil in the hole on theend that is not connected to anything. Use the pencil to move the strip aroundits hinge, drawing all the places where a 3-inch side could end.

d. Remove the connected strips from your paper.

2. What shape have you drawn while moving the 3-inch strip around? Why? Which toolin your geometry toolkit can do something similar?

3. Use your drawing to create two unique triangles, each with a base of length 4 inchesand a side of length 3 inches. Use a different color to draw each triangle.

4. Reposition the strips on the paper so that the 4-inch strip is on top of the 4-inchsegment again. In the 3-inch strip on the left side, put the tip of your pencil in the holeon the end that is not connected to anything. Use the pencil to move the strip aroundits hinge, drawing all the places where another 3-inch side could end.

5. Using a third color, draw a point where the two marks intersect. Using this thirdcolor, draw a triangle with side lengths of 4 inches, 3 inches, and 3 inches.

Unit 1 Lesson 15: Building Polygons 107

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Lesson 15 Summary

If we want to build a polygon with two given side lengths that share a vertex, we can thinkof them as being connected by a hinge that can be opened or closed:

All of the possible positions of the endpoint of the moving side form a circle:


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You may have noticed that sometimes it is not possible to build a polygon given a set oflengths. For example, if we have one really, really long segment and a bunch of shortsegments, we may not be able to connect them all up. Here's what happens if you try tomake a triangle with side lengths 21, 4, and 2:

The short sides don't seem like they can meet up because they are too far away from eachother.

If we draw circles of radius 4 and 2 on the endpoints of the side of length 21 to representpositions for the shorter sides, we can see that there are no places for the short sides thatwould allow them to meet up and form a triangle.

In general, the longest side length must be less than the sum of the other two side lengths.If not, we can’t make a triangle!

If we can make a triangle with three given side lengths, it turns out that the measures ofthe corresponding angles will always be the same. For example, if two triangles have sidelengths 3, 4, and 5, they will have the same corresponding angle measures.


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Lesson 15 Practice Problems1. A rectangle has side lengths of 6 units and 3 units. Could you make a quadrilateral

that is not identical using the same four side lengths? If so, describe it.

2. Come up with an example of three side lengths that can not possibly make a triangle,and explain how you know.

3. In the diagram, the length of segment is 10 units and the radius of the circlecentered at is 4 units. Use this to create two unique triangles, each with a side oflength 10 and a side of length 4. Label the sides that have length 10 and 4.

4. Select all the sets of three side lengths that will make a triangle.

A. 3, 4, 8

B. 7, 6, 12

C. 5, 11, 13

D. 4, 6, 12

E. 4, 6, 10


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5. Based on signal strength, a person knows their lost phone is exactly 47 feet from thenearest cell tower. The person is currently standing 23 feet from the same cell tower.What is the closest the phone could be to the person? What is the furthest theirphone could be from them?

6. Here is quadrilateral .

Draw the image of quadrilateral after each rotation using as center.

a. 90 degrees clockwise

b. 120 degrees clockwise

c. 30 degrees counterclockwise



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Lesson 16: Triangles with 3 Common Measures

Let’s contrast triangles.

16.1: 3 Sides; 3 AnglesExamine each set of triangles. What do you notice? What is the same about the triangles inthe set? What is different?

Set 1:

Set 2:


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16.2: 2 Sides and 1 AngleExamine this set of triangles.

1. What is the same about the triangles in the set? What is different?

2. How many different triangles are there? Explain or show your reasoning.

Unit 1Lesson 16: Triangles with 3 Common Measures

113

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16.3: 2 Angles and 1 SideExamine this set of triangles.

1. What is the same about the triangles in the set? What is different?

2. How many different triangles are there? Explain or show your reasoning.


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Lesson 16 Summary

Both of these quadrilaterals have a right angle and side lengths 4 and 5:

However, in one case, the right angle is between the two given side lengths; in the other, itis not.

If we create two triangles with three equal measures, but these measures are not next toeach other in the same order, that usually means the triangles are different. Here is anexample:


115

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Lesson 16 Practice Problems1. Are these two triangles identical? Explain how you know.

2. Are these triangles identical? Explain your reasoning.

3. Tyler claims that if two triangles each have a side length of 11 units and a side lengthof 8 units, and also an angle measuring , they must be identical to each other. Doyou agree? Explain your reasoning.


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4. a. Draw segment .

b. When is rotated around point , the resulting segment is the same as. Where could point be located?


5. Here is trapezoid .

Using rigid transformations on the trapezoid, build a pattern. Describe some of therigid transformations you used.



117

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Lesson 17: Drawing Triangles

Let’s see how many different triangles we can draw with certain measurements.

17.1: Using a Compass to Estimate Length1. Draw a angle.

2. Use a compass to make sure both sides of your angle have a length of 5 centimeters.

3. If you connect the ends of the sides you drew to make a triangle, is the third sidelonger or shorter than 5 centimeters? How can you use a compass to explain youranswer?


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17.2: How Many Can You Draw?1. Draw as many different triangles as you can with each of these sets of

measurements:

a. Two angles measure , and one side measures 4 cm.

b. Two angles measure , and one side measures 4 cm.

c. One angle measures , one angle measures , and one side measures 4 cm.

2. Which of these sets of measurements determine one unique triangle? Explain orshow your reasoning.

Unit 1 Lesson 17: Drawing Triangles 119

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In the diagram, 9 toothpicks are used to make three equilateral triangles. Figure out a wayto move only 3 of the toothpicks so that the diagram has exactly 5 equilateral triangles.

17.3: Revisiting How Many Can You Draw?1. Draw as many different triangles as you can with this set of measurements.

a. One angle measures , one side measures 4 cm, and one side measures 5 cm.

b. Do these measurements determine one unique triangle? How do you know?


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2. Draw as many different triangles as you can with each of these sets of anglemeasurements. Do either of these sets of measurements determine one uniquetriangle? Explain how do you know.

a. One angle measures , one measures , and one measures .

b. One angle measures , one measures , and one measures


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Using only a compass and the edge of a blank index card, draw a perfectly equilateraltriangle. (Note! The tools are part of the challenge! You may not use a protractor! You maynot use a ruler!)

Lesson 17 Summary

A triangle has six measures: three side lengths and three angle measures.

If we are given three measures, then sometimes, there is no triangle that can be made. Forexample, there is no triangle with side lengths 1, 2, 5, and there is no triangle with all threeangles measuring .

Sometimes, only one triangle can be made. By this we mean that any triangle we make willbe the same, having the same six measures. For example, if a triangle can be made withthree given side lengths, then the corresponding angles will have the same measures.


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Another example is shown here: an angle measuring between two side lengths of 6and 8 units. With this information, one unique triangle can be made.

Sometimes, two or more different triangles can be made with three given measures. Forexample, here are two different triangles that can be made with an angle measuringand side lengths 6 and 8. Notice the angle is not between the given sides.

Three pieces of information about a triangle’s side lengths and angle measures maydetermine no triangles, one unique triangle, or more than one triangle. It depends on theinformation.


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Lesson 17 Practice Problems1. Use a protractor to try to draw each triangle. Which of these three triangles is

impossible to draw?

a. A triangle where one angle measures and another angle measures

b. A triangle where one angle measures and another angle measures

c. A triangle where one angle measures and another angle measures

2. A triangle has an angle measuring , an angle measuring , and a side that is 6units long. The 6-unit side is in between the and angles.

a. Sketch this triangle and label your sketch with the given measures.

b. How many unique triangles can you draw like this?


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3. A triangle has sides of length 7 cm, 4 cm, and 5 cm. How many unique triangles canbe drawn that fit that description? Explain or show your reasoning.

4. A triangle has one side that is 5 units long and an adjacent angle that measures .The two other angles in the triangle measure and . Complete the twodiagrams to create two different triangles with these measurements.

5. Is it possible to make a triangle that has angles measuring 90 degrees, 30 degrees,and 100 degrees? If so, draw an example. If not, explain your reasoning.


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Lesson 18: Rotate and Tessellate

Let's make complex patterns using transformations.

18.1: Deducing Angle MeasuresYour teacher will give you some shapes.

1. How many copies of the equilateral triangle can you fit together around a singlevertex, so that the triangles’ edges have no gaps or overlaps? What is the measure ofeach angle in these triangles?

2. What are the measures of the angles in the

a. square?

b. hexagon?

c. parallelogram?

d. right triangle?

e. octagon?

f. pentagon?


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18.2: Tessellate This1. Design your own tessellation. You will need to decide which shapes you want to use

and make copies. Remember that a tessellation is a repeating pattern that goes onforever to fill up the entire plane.

2. Find a partner and trade pictures. Describe a transformation of your partner’s picturethat takes the pattern to itself. How many different transformations can you find thattake the pattern to itself? Consider translations, reflections, and rotations.

3. If there’s time, color and decorate your tessellation.

18.3: Rotate That1. Make a design with rotational symmetry.

2. Find a partner who has also made a design. Exchange designs and find atransformation of your partner’s design that takes it to itself. Consider rotations,reflections, and translations.

3. If there’s time, color and decorate your design.

Glossary

tessellation•

Unit 1 Lesson 18: Rotate and Tessellate 127

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Learning TargetsLesson 1: Moving in the Plane

I can describe how a figure moves and turns to get from one position to another.

Lesson 2: Naming the Moves

I can identify corresponding points before and after a transformation.

I know the difference between translations, rotations, and reflections.

Lesson 3: Making the Moves

I can use grids to carry out transformations of figures.

I can use the terms translation, rotation, and reflection to precisely describetransformations.

Lesson 4: Coordinate Moves

I can apply transformations to points on a grid if I know their coordinates.

Lesson 5: Describing Transformations

I can apply transformations to a polygon on a grid if I know the coordinates of itsvertices.

Lesson 6: No Bending or Stretching

I can describe the effects of a rigid transformation on the lengths and angles in apolygon.

Lesson 7: Rotation Patterns

I can describe how to move one part of a figure to another using a rigidtransformation.

Lesson 8: Moves in Parallel

I can describe the effects of a rigid transformation on a pair of parallel lines.

If I have a pair of vertical angles and know the angle measure of one of them, I canfind the angle measure of the other.

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Lesson 9: Composing Figures

I can find missing side lengths or angle measures using properties of rigidtransformations.

Lesson 10: What Is the Same?

I can decide visually whether or not two figures are congruent.

Lesson 11: Congruence

I can decide using rigid transformations whether or not two figures are congruent.

I can use distances between points to decide if two figures are congruent.

Lesson 12: Alternate Interior Angles

I can find unknown angle measures by reasoning about complementary orsupplementary angles.

If I have two parallel lines cut by a transversal, I can identify alternate interior anglesand use that to find missing angle measurements.

Lesson 13: Adding the Angles in a Triangle

If I know two of the angle measures in a triangle, I can find the third angle measure.

Lesson 14: Parallel Lines and the Angles in a Triangle

I can explain using pictures why the sum of the angles in any triangle is 180 degrees.

Lesson 15: Building Polygons

I can show that the 3 side lengths that form a triangle cannot be rearranged to form adifferent triangle.

I can show that the 4 side lengths that form a quadrilateral can be rearranged toform different quadrilaterals.

I can show whether or not 3 side lengths will make a triangle.

Lesson 16: Triangles with 3 Common Measures

I understand that changing which sides and angles are next to each other can makedifferent triangles.

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Unit 1 Learning Targets 129

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Lesson 17: Drawing Triangles

Given two angle measures and one side length, I can draw different triangles withthese measurements or show that these measurements determine one uniquetriangle or no triangle.

Lesson 18: Rotate and Tessellate

I can repeatedly use rigid transformations to make interesting repeating patterns offigures.

I can use properties of angle sums to reason about how figures will fit together.

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Glossaryalternate interior anglesInterior angles are angles that are made by a transversal crossing two parallel lines. Theyare the angles that lie between the parallel lines, not outside them.

If two interior angles lie on opposite sides of thetransversal they are called alternate interior angles.

In the figure, and are alternate interior angles, andand are also alternate interior angles.

clockwiseClockwise means to turn in the same direction as the handsof a clock. The top turns to the right. This diagram showsFigure A turned clockwise to make Figure B.

complementaryTwo angles are complementary to each other if their measures add up to . The twoacute angles in a right triangle are complementary to each other.

congruentOne figure is congruent to another if it can be moved with translations, rotations, andreflections to fit exactly over the other.

Unit 1 Glossary 131

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In the figure, Triangle A is congruent toTriangles B, C, and D. A translation takesTriangle A to Triangle B, a rotation takesTriangle B to Triangle C, and a reflectiontakes Triangle C to Triangle D.

coordinate planeThe coordinate plane is a system for tellingwhere points are. For example. point islocated at on the coordinate plane,because it is three units to the right and twounits up.

correspondingWhen part of an original figure matches up with part of a copy, we call themcorresponding parts. These could be points, segments, angles, or distances.

For example, point in the first trianglecorresponds to point in the secondtriangle. Segment corresponds tosegment .

counterclockwiseCounterclockwise means to turn opposite of the way the hands of a clock turn. Thetop turns to the left.


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This diagram shows Figure A turned counterclockwise to makeFigure B.

imageTranslations, rotations, and reflections move objects in the plane. Points, segments, andother parts of the original all have corresponding parts on the “moved object.” The movedobject is called the image.

For example, here is triangle and a translationto the right and up which is labeled .

Point in the image corresponds to point ,segment in the image corresponds to segment

, and angle corresponds to angle .

reflectionA reflection across a line moves every point on a figure to a point directly on the oppositeside of the line. The new point is the same distance from the line as it was in the originalfigure.

This diagram shows a reflection of A over linethat makes the mirror image B.

Unit 1 Glossary 133

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right angleWhen you divide a straight angle into two angles with equal measures, each of the twoangles is a right angle. For example, the four corners of a square are right angles.

rigid transformationA rigid transformation is a move that does not change any measurements of a figure.Translations, rotations, and reflections are rigid transformations, as is any sequence ofthese.

rotationA rotation moves every point on a figure around a center by a given angle in a specificdirection.

This diagram shows Triangle A rotatedaround center by 55 degrees clockwise toget Triangle B.

sequence of transformations

straight angleA straight angle is an angle that forms a straight line. It measures 180 degrees.

supplementaryTwo angles are supplementary to each other if their measuresadd up to .

For example, angle is supplementary to angle ,because they add up to a straight angle, which has measure

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tessellationA tessellation is a repeating pattern of one or more shapes. The sides of the shapes fittogether perfectly and do not overlap. The pattern goes on forever in all directions.

This diagram shows part of a tessellation.

transformationA transformation is a translation, rotation, reflection, or dilation, or combination of these.There is also a more general concept of a transformation of the plane that is not discussedin grade 8.

translationA translation moves every point in a figure a given distance in a given direction.

This diagram shows a translation of Figure A to Figure Busing the direction and distance given by the arrow.

transversal

vertexA vertex is a point where two or more edges meet. When wehave more than one vertex, we call them vertices.

The vertices in this polygon are labeled , , , , and .

Unit 1 Glossary 135

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vertical anglesVertical angles are opposite angles that share the same vertex. They are formed by a pairof intersecting lines. Their angle measures are equal.

For example, angles and arevertical angles. If angle measures ,then angle must also measure .

Angles and are another pair ofvertical angles.


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Notes

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accelerated sample1 - prek 12 · sample copyright im 2019. im 6–8 math was originally developed...

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